The present invention is directed to the field of reservoir engineering, and is particularly applicable to petroleum reservoirs which are heterogeneous due to variations of permeability and porosity of the rock formation, and viscosity and compressibility of flowing fluids.
One common method for determining characteristics of a composite reservoir involves injecting fluids into, or withdrawing fluids from, an active, vertical well in order to create a pressure disturbance. Using parameters obtained from various sources such as well logs, well tests, and laboratory analyses of core and fluid samples, engineers can determine characteristics of the reservoir.
One of the parameters used as part of the analysis of the results of well tests is the “radius of investigation,” also referred to as the “radius of drainage.” The radius of investigation is the distance away from the starting location (where the well is located), at any given time, of the peak of the pressure disturbance. The radius investigation is an important parameter. For example, in order to analyze composite reservoir systems, petroleum engineers often need to estimate the radius of investigation for one, or a combination of, the following reasons:                1. To estimate the volume of influence while running transient well tests;        2. To optimize durations of transient tests in terms of sampling designated volumes;        3. To determine minimum reserves of fluids, including hydrocarbons;        4. To estimate the time to stabilization for a given well spacing;        5. To calculate the time when interference between two neighboring wells will be encountered; and        6. To find the locations of new wells to be drilled, hence to determine well spacing.        
FIG. 1 shows schematically a radial composite reservoir, which has been divided into “n” radial regions located around the wellbore O. The boundaries between any two adjacent regions are designated with the corresponding radii. When a disturbance is created in the wellbore by initiating production and injection of fluid, i.e., at location O, a disturbance wave propagates radially outward toward the deep regions of the reservoir, from one concentric region to the next. The peak of the pressure disturbance is referred to as the “radius of investigation,” or alternatively as the “radius of drainage.” The radius of investigation is initially zero as long as there is no production or injection of fluid at the well. When a disturbance is created, the location of the radius of investigation moves radially outward at a rate dependent upon the isotropic properties of the rocks. Therefore, within the reservoir domain encircled by the location of the radius of investigation, there exists some detectable pressure gradient which affects the pressure behavior at the wellbore. The radius of investigation boundary expands with time and sweeps through the radially outward regions. FIG. 1 shows, for illustrative purposes, the location of the radius of investigation rij at time Δt (as indicated by one of the dotted circles).
Currently, petroleum engineers determine the radius of investigation by building a rigorous reservoir simulation model. Reservoir simulation models are reasonably accurate, but are time consuming and expensive to create and require developing software.
In “Well Test Analysis: The Use of Advanced Interpretation Models,” D. Bourdet, Elsevier Science B.V., p. 194 (2002), the author proposes a well test analysis method using a simple formula to estimate the radius of investigation at any given time. The simple formula used, however, is capable of considering variations in permeability only, and for such reason may result in large errors, as high as 100% or more in the event where the porosity and the total system compressibility vary from one region to another. Thus, the simple formula method is not reliable for everyday use.
Two publications, “Estimation of Reservoir Properties Using Transient Pressure Data,” D. W. Vasco, H. Keers, and K. Karasaki, Water Resources Research, Vol. 36, No. 12, pp. 3447-3465 (December 2000); and “A Streamline Approach for integrating Transient Pressure Data Into High-Resolution Reservoir Models,” K. N. Kulkarni, A. Datta-Gupta, and D. W. Vasco, SPE Journal, pp. 273-282 (September 2001), show that a high-frequency solution to the “diffusivity formula” demonstrates that, during a propagation test, the speed of propagation is a function of the rock and fluid properties in the reservoir. These studies indeed validate the mathematical equations used in this invention.